Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 13\cdot 79 + 27\cdot 79^{2} + 58\cdot 79^{3} + 50\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 18\cdot 79^{2} + 44\cdot 79^{3} + 65\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 53\cdot 79 + 59\cdot 79^{2} + 19\cdot 79^{3} + 34\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 36\cdot 79 + 64\cdot 79^{2} + 4\cdot 79^{3} + 61\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 + 55\cdot 79 + 6\cdot 79^{2} + 75\cdot 79^{3} + 11\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 + 56\cdot 79 + 36\cdot 79^{2} + 47\cdot 79^{3} + 2\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 57 + 65\cdot 79 + 72\cdot 79^{2} + 23\cdot 79^{3} + 7\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 61 + 35\cdot 79 + 30\cdot 79^{2} + 42\cdot 79^{3} + 3\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,6)(7,8)$ |
| $(1,7,3,8)(2,5,6,4)$ |
| $(1,2)(3,6)(4,8)(5,7)$ |
| $(1,3)(2,6)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,6)(4,5)(7,8)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,8)(5,7)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(2,6)(7,8)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,4)(3,8)(5,6)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,5,3,4)(2,7,6,8)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,4,3,5)(2,8,6,7)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,7,3,8)(2,5,6,4)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,6,3,2)(4,7,5,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,5,3,4)(2,8,6,7)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
